Non-splittability of the rational homology cobordism group of 3-manifolds

The smooth rational homology cobordism group of rational homology three spheres, T, contains subgroups T_p generated by 3-manifolds with first homology p-torsion, where p is a prime. Rochlin's theorem and gauge theoretic methods show that the inclusion of the direct sum of the T_p into T has infinitely generated kernel. We use Heegaard-Floer methods to show that the cokernel is infinitely generated. On the level of Witt groups of linking forms and conjecturally in the topological category, the inclusion is an isomorphism. One application is the demonstration of the failure in dimension three of an algebraic and higher dimensional theorem of Stoltzfus regarding primary splittings in the knot concordance group. We also build on work of the second author with Hedden and Ruberman to prove that the homology cobordism group of rational homology three spheres that bound topological rational homology 4-balls remains infinitely generated when modded out by the homology cobordism group of integral homology 3-spheres.